A&A 587, A48 (2016) Astronomy DOI: 10.1051/0004-6361/201527573 & c ESO 2016 Astrophysics

Asteroid models from the Lowell photometric database

J. Durechˇ 1, J. Hanuš2,3, D. Oszkiewicz4,andR.Vancoˇ

1 Astronomical Institute, Faculty of Mathematics and Physics, Charles University in Prague, V Holešovickáchˇ 2, 180 00 Prague 8, Czech Republic e-mail: [email protected] 2 Centre National d’Études Spatiales, 2 place Maurice Quentin, 75039 Paris Cedex 01, France 3 Laboratoire Lagrange, UMR 7293, Université de la Côte d’Azur, CNRS, Observatoire de la Côte d’Azur, Bd de l’Observatoire, CS 34229, 06304 Nice Cedex 04, France 4 Astronomical Observatory Institute, Faculty of Physics, A. Mickiewicz University, Słoneczna 36, 60-286 Poznan,´ Poland Received 16 October 2015 / Accepted 18 December 2015

ABSTRACT

Context. Information about shapes and spin states of individual is important for the study of the whole population. For asteroids from the main belt, most of the shape models available now have been reconstructed from disk-integrated photometry by the lightcurve inversion method. Aims. We want to significantly enlarge the current sample (∼350) of available asteroid models. Methods. We use the lightcurve inversion method to derive new shape models and spin states of asteroids from the sparse-in-time photometry compiled in the Lowell Photometric Database. To speed up the time-consuming process of scanning the period parameter space through the use of convex shape models, we use the distributed computing project Asteroids@home, running on the Berkeley Open Infrastructure for Network Computing (BOINC) platform. This way, the period-search interval is divided into hundreds of smaller intervals. These intervals are scanned separately by different volunteers and then joined together. We also use an alternative, faster, approach when searching the best-fit period by using a model of triaxial ellipsoid. By this, we can independently confirm periods found with convex models and also find rotation periods for some of those asteroids for which the convex-model approach gives too many solutions. Results. From the analysis of Lowell photometric data of the first 100 000 numbered asteroids, we derived 328 new models. This almost doubles the number of available models. We tested the reliability of our results by comparing models that were derived from purely Lowell data with those based on dense lightcurves, and we found that the rate of false-positive solutions is very low. We also present updated plots of the distribution of spin obliquities and pole longitudes that confirm previous findings about a non-uniform distribution of spin axes. However, the models reconstructed from noisy sparse data are heavily biased towards more elongated bodies with high lightcurve amplitudes. Conclusions. The Lowell Photometric Database is a rich and reliable source of information about the spin states of asteroids. We expect hundreds of other asteroid models for asteroids with numbers larger than 100 000 to be derivable from this data set. More models will be able to be reconstructed when Lowell data are merged with other photometry. Key words. minor planets, asteroids: general – methods: data analysis – techniques: photometric

1. Introduction shown that sparse photometry can be used to solve the lightcurve inversion problem and further simulations confirm this (Durechˇ Large all-sky surveys like Catalina, Pan-STARRS, etc. image et al. 2005, 2007). Afterwards, real sparse data were used either the sky every night to discover new asteroids and detect those alone or in combination with dense lightcurves and new asteroid that are potentially hazardous. The main output of these surveys models were derived (Durechˇ et al. 2009; Cellino et al. 2009; is a steadily increasing number of asteroids with known orbits. Hanuš et al. 2011, 2013c). The aim of these efforts was to de- Apart from astrometry that is used for orbit computation, these rive new unique models of asteroids, i.e., their sidereal rotation surveys also produce photometry of asteroids. This photome- periods, shapes, and direction of spin axis. try contains, in principle, information about asteroid rotation, shape, and surface properties. However, because of its poor qual- Another approach to utilize sparse data was to look for ity (when compared with a dedicated photometric measurements changes in the mean brightness as a function of the aspect angle, of a single asteroid) the signal corresponding to asteroid’s rota- which led to estimations of spin-axis longitudes for more than tion is usually drowned in noise and systematic errors. However, 350 000 asteroids (Bowell et al. 2014) from the so-called Lowell there have been recent attempts to use sparse-in-time photome- Observatory photometric database (Oszkiewicz et al. 2011). try to reconstruct the shape of asteroids. Kaasalainen (2004)has In this paper, we show that the Lowell photometric data set can also be used for solving the full inversion problem. By pro- Tables 1 and 2 are only available at the CDS via anonymous ftp to cessing Lowell photometry for the first 100 000 numbered as- cdsarc.u-strasbg.fr (130.79.128.5)orvia teroids, we derived new shapes and spin states for 328 aster- http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/587/A48 oids, which almost doubles the number of asteroids for which Czech National Team. the photometry-based physical model is known.

Article published by EDP Sciences A48, page 1 of 6 A&A 587, A48 (2016)

We describe the data, the inversion method, and the relia- 2.3. Asteroids@home bility tests in Sect. 2, the results in Sect. 3, and we conclude in Sect. 4. Asteroids@home is a volunteer-based computing project built on the Berkeley Open Infrastructure for Network Computing (BOINC) platform. Because the scanning of the period parameter space is the so-called embarrassingly parallel prob- 2. Method lem, we divided the whole interval of 2−100 h into smaller in- The lightcurve inversion method of Kaasalainen et al. (2001)that tervals (typically hundreds), which were searched individually we applied was reviewed by Kaasalainen et al. (2002)andmore on the computers of volunteers connected to the project. The units sent to volunteers had about the same CPU-time demand. recently by Durechˇ et al. (2016a). We used the same implemen- Results from volunteers were sent back to the BOINC server tation of the method as Hanuš et al. (2011), where the reader and validated. When all units belonging to one particular aster- is referred to for details. Here we describe only the general ap- oid were ready, they were connected and the global minimum proach and the details specific for our work. was found. The technical details of the project are described in Durechˇ et al. (2015) 2.1. Data As the data source, we used the Lowell Observatory photomet- 2.4. Ellipsoids ric database (Bowell et al. 2014). This is photometry provided To find the in sparse data, we also used an al- to Centre (MPC) by 11 of the largest surveys that ternative approach that was based on the triaxial ellipsoid shape were re-calibrated in the V-band using the accurate photometry model and a geometrical light-scattering model (Kaasalainen & of the Sloan Digital Sky Survey. Details about the data reduc- ˇ tion and calibration can be found in Oszkiewicz et al. (2011). Durech 2007). Its advantage is that it is much faster than using The data are available for about ∼330 000 asteroids. Typically, convex shapes because the brightness can be computed analyti- there are several hundreds of photometric points for each as- cally (it is proportional to the illuminated projected area, Ostro teroid. The length of the observing interval is ∼10–15 yr. The & Connelly 1984). On top of that, contrary to the convex mod- largest amount of data is for the low-numbered asteroids and de- elling, all shape models automatically fulfill the physical condi- tion of rotating along the principal axis with the largest momen- creases with increasing asteroid numbers. For example, the av- ffi erage number of data points is ∼480 for asteroids with number tum of inertia. The accuracy of this simplified model is su cient < ∼ > to reveal the correct rotation period as a significant minimum of 10 000 and 45 for those 300 000. The accuracy of the data χ2 is around 0.10−0.20 mag. in the period parameter space. That period is then used as a start point for the convex inversion for the final model. In many For each asteroid and of observation, we computed cases when the convex models gives many equally good solu- the asteroid-centric vectors towards the Sun and the Earth in tions with different periods, this method provides a unique and the Cartesian ecliptic coordinate frame – these were needed to correct rotation period. compute the illumination and viewing geometry in the inversion code. 2.5. Restricted period interval 2.2. Convex models As mentioned above, the interval for period search was 2–100 h. However, for many asteroids, their rotation period is known To derive asteroid models from the optical data, we used the from observations of their lightcurves. The largest database of lightcurve inversion method of Kaasalainen & Torppa (2001) asteroid rotation periods is the Lightcurve Asteroid Database and Kaasalainen et al. (2001), the same way as Hanuš et al. (LCDB) compiled by Warner et al. (2009) and regularly up- (2011). Essentially, we searched for the best-fit model by dated2. If we take information about the rotation period as an densely scanning the rotation period parameter space. We de- a priori constraint, we can narrow the interval of possible peri- − cided to search in the interval of 2 100 h. The lower limit ods and significantly shrink the parameter space. For this pur- roughly corresponds to the observed rotation limit of asteroids pose, we used only reliable period determinations from LCDB ∼ larger than 150 m (Pravec et al. 2002), the upper limit was set with quality codes U equal to 3, 3-, or 2+.However,evenfor arbitrarily to cover most of the rotation periods for asteroids de- these quality codes, the LCDB period can be wrong (for exam- termined so far. For each trial period, we started with ten initial ples see Marciniak et al. 2015) resulting in a wrong shape model. pole directions that were isotropically distributed on a sphere. For quality codes 3 and 3-, we restricted the search interval to This turned out to be enough not to miss any local minimum in P ± 0.05P,whereP was the rotation period reported in LCDB. the pole parameter space. In each period run, we recorded the pe- Similarly for U equal to 2+, we restricted the search interval to χ2 riod and value that correspond to the best fit. Then we looked P±0.1P. We applied this approach to both convex- and ellipsoid- χ2 for the global minimum of on the whole period interval and based period search. tested the uniqueness and stability of this globally best solution (see details in Sect. 2.6). For a typical data set, the number of trial periods is 2.6. Tests − 200 000 300 000, which takes about a month on one CPU. For each periodogram, there is formally one best model that cor- Because the number of asteroids we wanted to process was χ2 ∼ responds to the period with the lowest . However, the global 100 000, the only way to finish the computations in a reason- minimum in χ2 has to be significantly deeper than all other lo- able time was to use tens of thousands of CPUs. For this task, 1 cal minima to be considered as a reliable solution, rather than we used the distributed computing project Asteroids@home . just a random fluctuation. We could not use formal statistical

1 http://asteroidsathome.net 2 http://www.minorplanet.info/lightcurvedatabase.html

A48, page 2 of 6 J. Durechˇ et al.: Asteroid models from the Lowell photometric database tools to decide whether the lowest χ2 value is statistically signif- 50 icant or not, because the data were also affected by systematic errors. Instead, to select only robust models, we set up several 40 tests, which each model had to pass to be considered as a reli- able model. 30 2 1. The lowest χ corresponding to the rotation period Pmin is 2 at least 5% lower than all other χ values for periods outside Number 20 ± . 2 /Δ Δ the Pmin 0 5Pmin T interval, where T is the time span of observations (Kaasalainen 2004). The value of 5% was cho- 10 sen such that the number of unique models was as large as possible while keeping the number of false positive solutions ∼ 0 very low ( 1%). The comparison was done with respect to 0 102030405060 models in DAMIT (see Sect. 3.1). Difference between pole directions [deg] 2. When using convex models for scanning the period parame- ter space, we ran the period search for two resolutions of the Fig. 1. Histogram of differences between the pole directions of models convex model – the degree and order of the harmonics series derived from Lowell data and those archived in DAMIT. expansion that parametrized the shape was three or six. The periods Pmin corresponding to these two resolutions had to Asteroid Models from Inversion Techniques (DAMIT3, Durechˇ agree within their errors (and both had to pass the test nr. 1). > et al. 2010). For this subset, we could compare our results from 3. Because we realized that Pmin ∼ 20 h often produced an inversion of Lowell data with independent models (assumed false positive solutions, we accepted only models with Pmin to be reliable) from DAMIT. In total, there were 279 models shorter than 20 h (when there was no information about the in DAMIT for comparison. For these models, we computed the rotation period from LCDB). difference between the DAMIT and Lowell rotation periods and 4. For a given Pmin, there are no more than two distinct (farther also the difference between the pole directions. Out of this set, ◦ χ2 than 30 apart) pole solutions with at least 5% deeper than almost all (275 models) have the same rotation periods (within other poles. the uncertainties) and the pole differences <50◦ of arc. The his- 5. Because of the geometry limited close to the ecliptic plane, togram of pole differences between DAMIT and our models is β two models that have the same pole latitudes and pole lon- shown in Fig. 1. Although there are some asteroids for which we λ ff ◦ gitudes that are di erent by 180 provide the same fit to got differences as large as ∼40−50◦, the mean value is 15◦ and disk-integrated data, and they cannot be distinguished from the median 13◦, which can be interpreted as a typical error in the each other (Kaasalainen & Lamberg 2006). Therefore we ac- pole determination that was based on Lowell data, assuming that cepted only such solutions that fulfilled the condition that if the poles from DAMIT have smaller errors (typically 5−10◦). there were two pole directions (λ1,β1)and(λ2,β2), the differ- ff ◦ As an example of the di erence between shape models, we ence in ecliptic latitudes |β1 − β2| has to be less than 50 and ◦ show results for asteroid (63) Ausonia. In Fig. 2, we compare the difference in ecliptic longitudes mod (|λ1 − λ2|, 180 ) ◦ our shape model, which we derived from Lowell sparse photom- has to be larger than 120 . etry, with that obtained by inversion of dense lightcurves (Torppa 6. The ratio of the moment of inertia along the principal axis et al. 2003). In general, the shapes derived from sparse photom- to that along the actual rotation axis should be less than 1.1. etry are more angular than those derived from dense lightcurves Otherwise the model is too elongated along the direction of and often have artificial sharp edges. the rotation axis and it is not considered a realistic shape. The four asteroids (5) Astraea, (367) Amicita, 7. For each asteroid that passed the above test, we created a (540) Rosamunde, and (4954) Eric, for which we got dif- bootstrapped lightcurve data set by randomly selecting the ferent solutions to DAMIT, are discussed below. We also same number of observations from the original data set. This discuss the five asteroids – (1753) Mieke, (2425) Shenzen, new data set was processed the same way as the original (6166) Univsima, (11958) Galiani, and (12753) Povenmire – for one (using either convex shapes or ellipsoids for the period which there is no model in DAMIT, but the period we derived search) and the model was considered stable only if the best- from the Lowell data does not agree with the data in LCDB. fit period Pmin from the bootstrapped data agreed with that from the original data. 8. We also visually checked all shape models, periodograms, (5) Astraea. From Lowell data, we got two pole directions ◦ ◦ ◦ ◦ and fits to the data to be sure that the shape model looked (λ, β) = (121 , −20 ) and (296 , −15 ), the former being about ◦ realistic and that there were no clear problems with the data 60 away from the DAMIT model of Hanuš et al. (2013b) with ◦ ◦ and residuals. In some rare cases we rejected models that the pole (126 , 40 ). The DAMIT model agrees with the adap- formally fitted the data, passed all the test, but were unreal- tive optics data as well as with the occultation silhouette from istically elongated or flat. 2008 and it is not clear why there is so large a difference in the pole direction, while the rotation periods are the same and the number of Lowell photometric points is also large (447 points).

3. Results (367) Amicita. The model derived from Lowell data has two ◦ ◦ ◦ ◦ 3.1. Comparison with independent models pole solutions (17 , −52 ) and (194 , −45 ) and rotation period of 5.05578h, while the DAMIT model of Hanuš et al. (2011)has From all ∼600 models that successfully passed the tests de- prograde rotation with poles of (21◦, 32◦) and (203◦, 38◦), with a scribedinSect.2.6, some were already modeled from other photometric data and the models were stored in the Database of 3 http://astro.troja.mff.cuni.cz/projects/asteroids3D

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(11958) Galiani. This asteroid was observed by Clark (2014), who determined the period 9.8013 ± 0.0023 h, which does not agree with our value of 8.24720 h. The reason is not clear, be- cause the data of Clark (2014) seem to fit this period correctly. We do not see any significant minimum in χ2 near 9.8 h in the periodogram.

(12753) Povenmire. The period of 12.854 h reported in the LCDB is based on the observations of Gary (2004). However, according to the same author5, the correct rotation period that is based on observations from 2010 is 17.5752 ± 0.0008 h, which agrees with our value. In summary, the frequency of false positive solutions that pass all reliability tests seems to be sufficiently low, around a few percent. However, the sample of models in DAMIT that we use for comparison is itself biased against low-amplitude long- period asteroids (Marciniak et al. 2015), so the real number of Fig. 2. Comparison between the shape model of (63) Ausonia recon- false positive solutions might be higher. structed from Lowell sparse data (top) and from dense lightcurves (bottom). 3.2. New models After applying all the tests described in Sect. 2.6, we selected only those asteroids with no model in DAMIT for publication. significantly different period of 5.05502 h. However, the DAMIT These are listed in Tables 1 (models from full interval 2−100 h) model is based on sparse data from US Naval Observatory and and 2 (models derived from a restricted period interval). The ta- Catalina and only two pieces of lightcurve by Wisniewski et al. bles list the pole direction(s) (one or two models), the sidereal ro- (1997) and it might not be correct. tation period (with uncertainty corresponding to the order of the last decimal place). The C/E code means the method by which Pmin was found – convex models (C) or ellipsoids (E). In some (540) Rosamunde. Although the periodogram obtained with cases, both methods independently gave the same value of Pmin the convex model approach shows a minimum for 9.34780 h – (then CE code). All new shape models and the photometric data the same as the DAMIT model of Hanuš et al. (2013a) – this areavailableinDAMIT. minimum was not deep enough to pass the test nr. 1. However, Forsomeoftheseasteroids,Hanuš et al. (2016) obtained the second-best minimum for a convex model at 7.82166 h ap- independent models by applying the same lightcurve inver- peared as the best solution for the ellipsoid approach and passed sion method on sparse data, which they combined with dense all tests leading to a wrong model. lightcurves. These asteroids (not yet published in DAMIT) are marked by asterisk in the Tables 1 and 2. For all of them (56 in total), our rotation periods agree with those of Hanuš et al. (4954) Eric. The DAMIT model of Hanuš et al. (2013c)hasa ff pole direction of (86◦, −55◦), which is almost exactly opposite to (2016) within their uncertainties and pole directions di er by our value of (261◦, 70◦). Moreover, even the rotation periods are 10–20 degrees on average. By way of comparison, this is a sim- different by about 0.0003 h, which is more than the uncertainty ilar result to the DAMIT models (Sect. 3.1,Fig.1) and it inde- interval. pendently confirms the reliability of our models based on only Lowell data.

(1753) Mieke. The rotation period of 8.9 h was determined by 3.3. Statistics of pole directions Lagerkvist (1978) from two (1.5 and 5 h) noisy lightcurves. Given the quality of the data, this period is not in contradiction Together with models from DAMIT, we now have a sample of with our value of 10.19942h. shape models for 717 asteroids (685 MBAs, 13 NEAs, 10 Mars- crossers, 7 Hungarias, 1 Hilda, and 1 Trojan). The statistical analysis of the pole distribution confirms the previous find- . ± . (2425) Shenzen. The rotation period of 14 715 0 012 h ings. Namely, the distribution of spin directions is not isotropic was determined by Hawkins & Ditteon (2008). Our value of (Kryszczynska´ et al. 2007). Moreover, the distribution of pole / 9.83818 h is close to 2 3 of their. In the periodogram, there is obliquities (an angle between the spin vector and the normal no significant minimum around 14.7 h. to the orbital plane) depends on the size of an asteroid. We plot the dependence of obliquity on the size in Fig. 3 for main- belt (MBAs) and near-Earth (NEAs) asteroids. There is a clear (6166) Univsima. The lightcurve is published online in the database of R. Behrend4. However, the period of 9.6 h is based trend of smaller asteroids clustering towards extreme values of on only 12 points, which covers about half of the reported pe- obliquity. This was explained by Hanuš et al. (2011)asYORP- induced evolution of spins (Hanuš et al. 2013c). The distribu- riod, so we think that this preliminary result is not in contradic- ◦ tion with our period of ∼11.4h. tion of obliquities is not symmetric around 90 (Fig. 4). As no- ticed by Hanuš et al. (2013c), the retrograde rotators are more

4 http://obswww.unige.ch/~behrend/page5cou.html 5 http://brucegary.net/POVENMIRE/

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MBAs 70 0 45 60

[deg] 60 ε 50 90

120 40 135

Pole obliquity 180 30 1 10 100 Size [km] 20 NEAs Number of asteroids 0 10 45

[deg] 60 ε 0 0 50 100 150 90 Pole longitude mod(λ, 180) [deg] 120 λ 135 Fig. 5. Histograms of the distribution of pole longitude for 685 main-

Pole obliquity 180 belt asteroids. 1 10 100 Size [km] real data from the Lowell database. Therefore, we postpone this ε Fig. 3. Distribution of pole obliquity as a function of size for investigation for a future paper. 575 main-belt and 13 near-Earth asteroids. For near-Earth asteroids, the excess of retrograde rotators can be explained by the Yarkovsky-induced delivery mechanism D < 30 km D > 60 km 150 20 from the main belt through resonances (La Spina et al. 2004), although the number of NEA models in our sample is too small 15 for any reliable statistics. 100 We also see a clear deviation from a uniform distribution of 10 pole longitudes in Fig. 5. Because of ambiguity in pole direc- tion (often there are two solutions with similar latitudes and the 50 ◦ 5 difference in longitudes of about 180 ), we plotted the distribu- ◦

Number of asteroids Number of asteroids tion modulo 180 . The histogram shows an excess of longitudes 0 0 around 50−100◦. This was already announced by Bowell et al. −1 0 1 −1 0 1 cos ε cos ε (2014), who processed the Lowell data set using a different ap- proach, estimated spin-axis longitudes for more than 350 000 as- Fig. 4. Histograms of the distribution of pole obliquities ε for asteroids teroids, and revealed an excess of longitudes at 30−110◦ and a with diameters <30 km and >60 km, respectively. paucity at 120−180◦. The explanation of this phenomenon re- mains unclear. concentrated to −90◦, probably because prograde rotators are af- fected by resonances. For larger asteroids, there is an excess of 4. Conclusions prograde rotators that might be primordial (Kryszczynska´ et al. 2007; Johansen & Lacerda 2010). The new models presented in this paper significantly enlarge the However, the current sample of asteroid models is far from sample of asteroids for which their spin axis direction and ap- being representative of the whole asteroid population. Because proximate shape are known. Because these models are based on the period search in sparse data is strongly dependent on the a limited number of data points, the shapes have to be interpreted lightcurve amplitude – the larger the amplitude the easier is to as only approximations of the real shapes of asteroids. Also the detect the correct rotation period in noisy data – more elongated pole directions need to be refined with more data if one is in- asteroids are reconstructed more easily than spherical ones. That terested in a particular asteroid. However, as an ensemble, the is why almost all the asteroids listed in Tables 1 and 2 have large models can be used in future statistical studies of asteroid spins, amplidudes of ∼>0.3 mag. The lightcurve inversion (based mostly for example. or exclusively on sparse data) is also less efficient for asteroids We believe that this is only the beginning of a with poles close to the ecliptic plane because, during some ap- production of shape and spin models from sparse photome- paritions, we observe them almost pole-on, thus with very small try. Although the number of models derivable from the Lowell amplitudes. This bias in the method was estimated to be of the Observatory photometric database is small compared to the total order of several tens percent (Hanuš et al. 2011). A much higher number of asteroids, the potential of Lowell photometry con- discrepancy (factor 3–4) in the successfully recovered pole di- sists in its combination with other data. Even a priori informa- rections between poles close-to and perpendicular-to the eclip- tion about the rotation period shrinks the parameter space that tic was found by Santana-Ros et al. (2015). But even such a has to be scanned, and a local minimum in a large parameters large selection effect cannot fully explain the significant “gap” space becomes a global minimum on a restricted interval. Of for obliquities between 60–120◦. To clearly show how the unbi- course, the reliability of this type of model depends critically ased distribution of pole obliquities looks like, we would have to on the reliability of the period. Lowell photometry can be com- carry out an extensive simulation on a synthetic population with bined with dense lightcurves that constrain the rotation period. realistic systematic and random errors to see the bias that is in- This way, models for about 250 asteroids were derived recently duced by the method, shape, and geometry. This sort of simula- by Hanuš et al. (2016), some of which confirm the models pre- tion would be more computationally demanding than processing sented in this paper. The database of asteroid rotation periods has

A48, page 5 of 6 A&A 587, A48 (2016) been increased dramatically by Waszczak et al. (2015)–their Durech,ˇ J., Scheirich, P., Kaasalainen, M., et al. 2007, in Near Earth Objects, our data can also be combined with Lowell photometry, and we ex- Celestial Neighbors: Opportunity and Risk, eds. A. Milani, G. B. Valsecchi, pect that other hundreds of models will be reconstructed from & D. Vokrouhlický (Cambridge: Cambridge University Press), 191 Durech,ˇ J., Kaasalainen, M., Warner, B. D., et al. 2009, A&A, 493, 291 this data set. Another promising approach is the combination of Durech,ˇ J., Sidorin, V., & Kaasalainen, M. 2010, A&A, 513, A46 sparse photometry with data from the Wide-field Infrared Survey Durech,ˇ J., Hanuš, J., & Vanco,ˇ R. 2015, Astron. Comp., 13, 80 Explorer (WISE) mission (Wright et al. 2010). Although WISE Durech,ˇ J., Carry, B., Delbo, M., Kaasalainen, M., & Viikinkoski, M. 2016a, in data were observed in mid-infrared wavelengths, Durechˇ et al. Asteroids IV, eds. P. Michel, F. DeMeo, & W. Bottke (Tucson: University of Arizona Press), in press (2016b) showed that thermally emitted flux can be treated as re- Durech,ˇ J., Hanuš, J., Ali-Lagoa, V., Delbo, M., & Oszkiewicz, D. 2016b, flected light to derive the correct rotation period and the shape in Proc. IAU Symp. 318, eds. S. Chesley, R. Jedicke, A. Morbidelli, & and spin model. This opens up a new possibility, because both D. Farnocchia (Cambridge: Cambridge University Press) Lowell and WISE data are available for tens of thousands of Gary, B. L. 2004, Minor Planet Bull., 31, 56 Hanuš, J., Durech,ˇ J., Brož, M., et al. 2011, A&A, 530, A134 asteroids. Hanuš, J., Brož, M., Durech, J., et al. 2013a, A&A, 559, A134 In general, the combination of more data sources is always Hanuš, J., Marchis, F., & Durech,ˇ J. 2013b, Icarus, 226, 1045 better than using them separately. By using Lowell photometry Hanuš, J., Durech,ˇ J., Brož, M., et al. 2013c, A&A, 551, A67 with dense lightcurves, WISE data, photometry from Gaia,etc., Hanuš, J., Durech,ˇ J., Oszkiewicz, D. A., et al. 2016, A&A, 586, A108 the number of available models will increase and the statistical Hawkins, S., & Ditteon, R. 2008, Minor Planet Bull., 35, 1 Johansen, A., & Lacerda, P. 2010, MNRAS, 404, 475 studies of spin and shape distribution will become more robust, Kaasalainen, M. 2004, A&A, 422, L39 being based on larger sets of models. Nevertheless, any infer- Kaasalainen, M., & Durech,ˇ J. 2007, in Near Earth Objects, our Celestial ence based on the models derived from lightcurves (and sparse Neighbors: Opportunity and Risk, eds. A. Milani, G. B. Valsecchi, & lightcurves in particular) has to take into account that the sam- D. Vokrouhlický (Cambridge: Cambridge University Press), 151 Kaasalainen, M., & Lamberg, L. 2006, Inverse Problems, 22, 749 ple of models is biased against more spherical shapes with low Kaasalainen, M., & Torppa, J. 2001, Icarus, 153, 24 lightcurve amplitudes and poles near the plane of ecliptic. Kaasalainen, M., Torppa, J., & Muinonen, K. 2001, Icarus, 153, 37 Kaasalainen, M., Mottola, S., & Fulchignomi, M. 2002, in Asteroids III, eds. Acknowledgements. We would not be able to process data for hundreds of W. F. Bottke, A. Cellino, P. Paolicchi, & R. P. Binzel (Tucson: University of thousands of asteroids without the help of tens of thousand of volunteers who Arizona Press), 139 joined the Asteroids@home BOINC project and provided computing resources Kryszczynska,´ A., La Spina, A., Paolicchi, P., et al. 2007, Icarus, 192, 223 from their computers. We greatly appreciate their contribution. 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